Free loop

In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a distinguished point. Formally, let $X$ be a topological space. Then a free loop in $X$ is an equivalence class of continuous functions from the circle $S^1$ to $X$. Two loops are equivalent if they differ by a reparameterization of the circle. That is, $f \sim g$ if there exists a homeomorphism $\psi : S^1 \rightarrow S^1$ such that $g = f\circ\psi$.

Thus, a free loop, as opposed to a based loop used in the definition of the fundamental group, is a map from the circle to the space without the basepoint-preserving restriction. Free homotopy classes of free loops correspond to conjugacy classes in the fundamental group.

In the last years, interest in the space of all free loops $LX$ has grown with the advent of string topology, i.e. the study of new algebraic structures on the homology of the free loop space.

See also

• Loop space
• Loop (topology)
• Quasigroup

Further reading

• Brylinski, Jean-Luc: Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008.
• Cohen and Voronov: Notes on String Topology